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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 8 — Apr. 12, 2010
  • pp: 7872–7885
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Optomechanical zipper cavity lasers: theoretical analysis of tuning range and stability

Thiago P. Mayer Alegre, Raviv Perahia, and Oskar Painter  »View Author Affiliations


Optics Express, Vol. 18, Issue 8, pp. 7872-7885 (2010)
http://dx.doi.org/10.1364/OE.18.007872


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Abstract

The design of highly wavelength tunable semiconductor laser structures is presented. The system is based on a one dimensional photonic crystal cavity consisting of two patterned, doubly-clamped nanobeams, otherwise known as a “zipper” cavity. Zipper cavities are highly dispersive with respect to the gap between nanobeams in which extremely strong radiation pressure forces exist. Schemes for controlling the zipper cavity wavelength both optically and electrically are presented. Tuning ranges as high as 75 nm are achieved for a nominal design wavelength of λ = 1.3 μm. Sensitivity of the mechanically compliant laser structure to thermal noise is considered, and it is found that dynamic back-action of radiation pressure in the form of an optical or electrical spring can be used to stabilize the laser frequency. Fabrication of zipper cavity laser structures in GaAs material with embedded self-assembled InAs quantum dots is presented, along with measurements of photoluminescence spectroscopy of the zipper cavity modes.

© 2010 Optical Society of America

1. Introduction

The tuning functionality of semiconductor laser cavities is an important attribute for a variety of applications, from spectroscopy [1–3

1. T. M. Hard, “Laser wavelength selection and output coupling by a grating,” Appl. Opt. 9, 1825–1830 (1970). [CrossRef] [PubMed]

] and lightwave communication [4

4. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers. New York, NY: Van Nostrand Reinhold, (1993).

], to cavity quantum-electro-dynamics (QED) studies of light matter interactions [5

5. K. Srinivasan and O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature 450, 862–865 (2007). [CrossRef] [PubMed]

]. In this context, there has been great interest in extending both the tuning bandwidth and the tuning speed [6

6. E. Bruce, “Tunable Lasers,” IEEE Spectrum 39, 35–39 (2002). [CrossRef]

, 7

7. A. Q. Liu and X. M. Zhang, “A review of MEMS external-cavity tunable lasers,” J. Micromech. Microeng. 17, R1–R13 (2007). [CrossRef]

]. Injection-current tuning of semiconductor lasers [8

8. L. Coldren and S. Corzine, “Continuously-tunable single-frequency semiconductor lasers,” IEEE J. Quantum Electron. 23, 903–908 (1987). [CrossRef]

], through the refractive index dependence of the semiconductor on carrier concentration, can provide large bandwidth (GHz and beyond [9

9. Y. Matsui, H. Murai, S. Arahira, S. Kutsuzawa, and Y. Ogawa, “30-GHz bandwidth 1.55 μm strain-compensated InGaAlAs-InGaAsP MQW laser,” IEEE Photonics Technol. Lett. 9, 25–27 (1997). [CrossRef]

]) but has limited tuning range. Temperature tuning is also widely used for frequency stabilizing and trimming of lasers [10

10. M. Kitamura, M. Seki, M. Yamaguchi, I. Mito, K. Kobayashi, and T. Matsuoka, “High-power single-longitudinal-mode operation of 1.3 μm DFB-DC-PBH LD,” Electron. Lett. 19, 840–841 (1983). [CrossRef]

], but is limited in both range and speed. Novel optofluidic techniques have been demonstrated in semiconductor lasers in the mid-IR [11

11. L. Diehl, B. G. Lee, P. Behroozi, M. Loncar, M. A. Belkin, F. Capasso, T. Aellen, D. Hofstetter, M. Beck, and J. Faist, “Microfluidic tuning of distributed feedback quantum cascade lasers,” Opt. Express 14, 11660–11667 (2006). [CrossRef] [PubMed]

, 12

12. V. Moreau, R. Colombelli, R. Perahia, O. Painter, L. R. Wilson, and A. B. Krysa, “Proof-of-principle of surface detection with air-guided quantum cascade lasers,” Opt. Express 16, 8387–6396 (2008). [CrossRef]

], however, such techniques are again typically slow and involve integration of a somewhat cumbersome fluidic delivery system. Faster, more dramatic tuning mechanisms have been achieved through micro-electro-mechanical-systems (MEMS). For example, tuning of a mirror on top of vertical cavity surface emitting lasers [13

13. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nature Photon. 2, 180–184 (2008). [CrossRef]

], tuning of passive microdisks via MEMS actuators [14

14. M.-C. M. Lee and M. C. Wu, “MEMS-Actuated microdisk resonators with variable power coupling ratios,” IEEE Photon. Technol. Lett. 17, 1034–1036 (2005). [CrossRef]

], and electromechanical tuning of microdisks [15

15. K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” arXiv:0909.0082v3 [quant-ph], (2009).

]. More recently, there has been increased interest in cavity optomechanical systems, in which optical forces through various forms of radiation pressure are used to mechanically manipulate the optical cavity [16

16. T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-Action at the mesoscale,” Science 321, 1172–1176 (2008). [CrossRef] [PubMed]

]. Such systems have the dual property of being highly dispersive versus mechanical actuation, and can be implemented in guided wave nanostructures. In this article, we consider the properties of a photonic crystal (PC) “zipper” optomechanical cavity [17

17. J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystaloptomechanical cavity,” Opt. Express 17, 3802–3817 (2009). [CrossRef] [PubMed]

, 18

18. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009). [CrossRef] [PubMed]

] as it pertains to on-chip semiconductor lasers. In particular, we propose a master-slave cavity system for optical-force actuation of the zipper cavity, and explore the range of tuning and frequency stability in such mechanically compliant laser cavity structures. We also compare optical force tuning to more conventional electrostatic tuning methods in such cavity structures. Finally, initial photoluminescence measurements of fabricated zipper cavities in GaAs membranes containing an active region consisting of self-assembled InAs quantum dots is presented.

The zipper cavity structure considered in this work is shown in schematic form in Fig. 1(b). The zipper cavity is comprised of two doubly-clamped nanobeams in which a one dimensional PC pattern has been applied to the beams. Such an optical cavity is highly sensitive to the gap between the two nanobeams, resulting in extremely large dispersion of the cavity frequency with nanobeam gap and a corresponding large radiation pressure force (per intra-cavity photon) pulling the beams together. As an illustration, the optical frequency dispersion as a function of gap for a GaAs zipper cavity designed for operation at λ = 1310 nm (W = 400 nm (width), t =250 nm (thickness), ℓ=15 μm (total length)) is simulated via finite-element-method (FEM) and plotted in Fig. 1(c). The strength of the radiation pressure force can be quantified by the derivative of cavity frequency (ωc) versus nanobeam gap (xnb), gOM = dωc/dxnb, and is shown to exponentially increase with decreasing nanobeam gap. The large optomechanical coupling provided by zipper cavity structures clearly makes them very good candidates for tunable laser cavities. Along with their extremely small mass, on the order of picograms, the optomechanical coupling strength of zipper cavities is on the order of ωc/λc, equivalent to the exchange of photon momentum with the mechanical system every optical cycle. Fabry Pérot (FP) cavity systems [Fig. 1(a)] typically have a mass on the order of grams and an optomechanical strength that scales with the inverse of the round trip time of the cavity. For comparison an effective length scale may be defined (LOM) over which photon momentum is transferred from the light field to the mechanical system. In the case of a FP cavity LOM is equal to the physical cavity length (Lc), whereas in the zipper cavity LOM ~ λc.

Fig. 1. (a) Master-slave scheme for tuning of an optomechanical cavity. (b) Master zipper cavity. (c) Master capacitive MEMS actuator. (d) Frequency dispersion and optomechanical coupling coefficient (gOM) of a zipper cavity designed for λ = 1310 nm operation as a function of gap size.

This work will be divided into three main sections. In Sec. 2 we explore an all-optical master-slave zipper cavity system where different order optical cavity modes of the same structure are used. In Sec. 3 a capacitive MEMS actuator (master) is used to control the (slave) zipper cavity. An experimental demonstration of an active quantum dot based GaAs zipper cavity will be presented in Sec. 4. Finally, parallels and differences between optical and electrical control will be highlighted, including discussion of noise control using the optical spring effect and its electrical counterpart.

2. All-optical optomechanical system

Δωs=gOM(s)Qo(m)gOM(m)kMωm2Pd.
(1)

As one can see from Eq. (1), large tuning relies on optimizing the opto-mechanical coupling of both modes and minimizing kM. While reducing kM one must take care not to reduce the mechanical frequency below the intended frequency response of the tunable system. Zipper cavities typically have a motional mass around tens of picograms and optomechanical coupling of 10–100 GHz/nm yielding large tuning ranges. Additionally, the zipper cavity geometry is amenable to trimming of Ωm and kM by adjusting the flexibility of the supporting structure. An example of a floppy structure is shown in Fig. 2(c) where the addition of struts [19

19. R. M. Camacho, J. Chan, M. Eichenfield, and O. Painter, “Characterization of radiation pressure and thermal effects in a nanoscale optomechanical cavity,” Opt. Express 17, 15726–15735 (2009). [CrossRef] [PubMed]

] reduced the spring constant by an order of magnitude to kM ≈ 0.1 N/m for the inplane motion.

To optimize gOM for semiconductor zipper cavities we begin by gaining intuition from perturbation theory of moving boundary conditions [20

20. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002). [CrossRef]

, 21

21. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009). [CrossRef] [PubMed]

]:

gOM=ω02dA(ΔεE2+Δ(ε1)D2)dVE2
(2)

where Δε = ε1 - ε2, Δ(ε-1) = ε-11 - 1/ε-12, ε1 is the dielectric constant of the cavity, ε2 is the dielectric constant of the surrounding, and the integration takes place over the faces perpendicular to the displacement vector. One can therefore increase gOM by increasing the electric field at the boundaries normal to the direction of motion. Since the fundamental motion of these structures are inplane, i.e. the motion changes only the zipper gap size, an increase of the optomechanical coupling can be achieved by reducing both the width and gap of the waveguide beams, squeezing the field out towards the boundaries, without compromising the radiation limited Q. Initial fabrication of zipper cavities in 255 nm thick GaAs and InP membranes indicated that a gap of xnb = 80 nm is a reasonable gap to achieve. Optical FEM simulations of the first order (slave) bonded defect mode [Fig. 2(b)] at λ = 1310 nm are carried out as a function of beam width (W = 300-675 nm). The nanobeams are t = 255 nm thick with an approximate refractive index n = 3.4. The PC mirror section of the cavity has a lattice constant a/λ = 0.254 and fill fraction (hy/a = 0.35, hx/W = 0.60). The parabolic defect region is 15 holes wide with a maximum lattice constant reduction of 15%. The Q and gOM, calculated via the perturbation theory of Eq. (2), are plotted as a function of W in Fig. 2(a). We choose a beam width W = 400 nm that maintains a high radiation limited Q (s) ≈ 107 and high optomechanical coupling g(s)OM = 94 GHz/nm. For this geometry we calculate a g(m)OM = 122 GHz/nm and Q(m) ≈ 5 × 106 for the 3rd order master mode at λ = 1388 nm [Fig. 2(b)].

Optical FEM simulations are then used to generate a cavity dispersion curve, for both modes, as a function of gap. A fit of the dispersion curves to a simple exponential model ω(xnb) = (ω0 - ωf)e -αxnb + ωf is then used to describe the gap-dependence of the optomechanical coupling. In this formula α is a decay constant describing the exponential sensitivity of the light field in the gap, ω0 is the cavity frequency when the beams are touching, and ωf is the cavity frequency when the beams are far apart and do not interact. Optomechanical coupling can then be modeled as gOM(xnb) = gOM(0)exnb. A plot of ω(xnb), the fit, and gOM(xnb) for the slave mode is plotted in Fig. 1(d). From this, the slave mode tuning can now be calculated as a function of the dropped optical power into the master mode, Pd, using Eq. 1. As the mechanical spring constant of zipper cavities can be tailored, we choose several representative values (kM = 0.1, 1.0, and 10.0 N/m) to simulate throughout this work which encompass experimentally feasible systems. Tuning of the slave mode versus master Pd is calculated for three mechanical spring constants and shown in Fig. 3(a). Since the applied optical force, for a constant dropped optical power, is nonlinear (exponential) with nanobeam gap, there is a position at which any additional optical force cannot be made up for by the restoring mechanical spring force, and no stable equilibrium can be achieved without the nanobeams touching. This position, for the minimum allowed slot gap (xnb,min), is known in the MEMS community as the in-use stiction point [22

22. N. Tas, T. Sonnenberg, H. Jansen, R. Legtenberg, and M. Elwenspoek, “Stiction in surface micromachining,” J. Micromech. Microeng. 6, 385–397 (1996). [CrossRef]

, 23

23. R. Maboudian and R. T. Howe, “Critical review: Adhesion in surface micromechanical structures,” J. Vac. Sci. Technol. B 15, 1–20 (1997). [CrossRef]

]. An analytical expression for minimum gap before stiction can be found, xnb,min = xnb,0-1, where xnb,0 is the fabricated (initial) nanobeam gap, as long as the optomechanical coupling has the above simple exponential behavior and one assumes the dropped optical power is independent of gap. Under such circumstances, a maximum allowed dropped power for the master mode is equal to Pmaxd = kMω2m/(2Q(m)og(m)OM(0)αexnb,min), resulting in a maximum tuning range of the slave mode for the geometry considered here with kM = 0.1 N/m of Δλ = 25 nm. For this spring constant the maximum power required to obtain the full tuning range is only Pd = 58 μW. In some experimental realizations a more nuanced analysis of the stiction point is necessary. For example, when a pump laser is used, an increase in pump power leads to cavity laser detuning. The maximum tuning range presented herein is therefore a conservative estimate and applies to situations where a wide band pump source such as a diode can be used.

Fig. 2. (a) Q calculated via FEM simulations and optomechanical coupling (gOM) calculated via FEM simulations and perturbation theory are plotted as a function of beam width (W) for the slave cavity at λ = 1310 nm. (b) FEM simulated electric field component (Ey) for the 1st (slave) and 3rd (master) order modes at λ = 1310 nm and λ = 1388 nm respectively. (c) Mechanical FEM simulation of inplane motion coupling to the optical modes (fmech = 600 kHz, meff = 7.15 pg, kM = 0.1 N/m). (d) Schematic of master-slave all optomechanical cavity system in the context of material with gain.
Fig. 3. (a) Slave mode tuning as a function of dropped pump power (Pd) into the master cavity mode for three distinct mechanical spring constants (k = 0.1,1.0 and 10 N/m). (b) Standard deviation of wavelength (σλ) due to thermal fluctuations at T = 300 K (room temperature) of the slave mode as a function of Pd into the master mode. Solid (dashed) lines are for Γt/2 (zero) detuning from the master cavity resonance. The gray dashed lines indicate the minimum wavelength deviation before the nanobeams stiction point.

Obviously, less stiff structures allow for lower power actuation; however, along with smaller spring constant comes a susceptibility to thermal Brownian motion. We estimate the standard deviation of the resonant wavelength (σλ) of the slave laser cavity by relating the position variance to the temperature and spring constant [24

24. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nature Photon. 3, 478–483 (2009). [CrossRef]

]. As can be seen from Fig. 3(b), for a kM =0.1 N/m device (dashed line) the wavelength noise will increase from σλ = 60 pm to σλ = 100 pm by the time its tuned the full range. The thermal Brownian motion can be significantly decreased using the optical spring effect [25

25. T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express 15, 17172–17205 (2007). [CrossRef] [PubMed]

,26

26. V. B. Braginsky, F. Y. Khalili, and K. S. Thorne, Quantum Measurement. Cambridge University Press, (1995).

] of the master mode. In the sideband unresolved regime, the in-phase component of the optical force leads to a modified effective dynamic spring constant [18

18. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009). [CrossRef] [PubMed]

, 24

24. J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nature Photon. 3, 478–483 (2009). [CrossRef]

, 27

27. Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009). [CrossRef] [PubMed]

]:

kM=kM+2gOM2PdΔωcΓt[Δ2+(Γt/2)2],
(3)

where Δ = ωl - ωc is the detuning of the input laser (ωl) from the cavity resonance (ωc) and Γt is the loaded optical cavity loss rate. The minimum frequency noise occurs for a maximum optical spring effect achieved when Δ = Γt/2. A comparison of wavelength noise with and without the optical spring effect for optimal detuning is shown on Fig. 3(b). At maximum tuning, for kM = 0.1 N/m, the increase in the dynamic spring constant decreases the thermal noise from ~ 100 pm to ~ 9 pm.

3. Electro-optomechanical system

The MEMS structure can be modeled as a parallel plate capacitor with one of the plates coupled to a spring [inset on Fig. 4(b)]. The capacitance for the MEMS structure as a function of the gap size between struts is simulated via FEM and plotted in Fig. 4(b). Since the capacitance of the structure is on the order of femtofarads (fF), when the structure is experimentally implemented, an appropriately designed RLC tank circuit will be used to reduce spurious impedances (Fig. 4(a) [34

34. M. A. Sillanpaa, J. Sarkar, J. Sulkko, J. Muhonen, and P. J. Hakonen, “Accessing nanomechanical resonators via a fast microwave circuit,” Appl. Phys. Lett. 95, 011909 (2009). [CrossRef]

]). For a DC voltage (Vdc), the applied attractive force on each strut as a function of the capacitor gap size (xcap) can be evaluated as Fel = (1/2) (dC/dxcap)V2dc. The total displacement, proportional to force, will produce a change in the optical frequency of the slave cavity through the change in capacitance. While the displacement does not depend on resistance, it is illustrative for comparison with the all-optical case (Sec. 2) to define the capacitive-mechanical coupling as gCM = dΓRC/dxcap, where ΓRC = 1/RC is the capacitor charging time constant. The tuning of the slave cavity is then given by:

Δωs=gOM(s)2gCM(m)kMΓRC2Vdc2R,
(4)

From Fig. 4(b) we can see that gCM is non-linearly dependent upon capacitor gap. As a result, we once again have a geometry-dependent relation between the minimum gap size and the maximum allowed voltage, i.e. the capacitor pull-in voltage. Using an empirical model for the capacitance C = a×x-bcap + C0, one can evaluate the minimal gap size as xcap, min = (b+1)/(b+2)xcap,0, where xcap,0 is the initial capacitor gap size and a and b are positive and real parameters found by fitting the empirical model to the FEM simulations [Fig. 4(b)]. This relation limits the maximum allowed voltage to be:

Vmax=[((b+1)b+1(b+2)b+2)(kMab)xcap,0(b+2)]1/2.
(5)

For a geometry where the initial nanobeam gap is xnb,0 = 80 nm and the initial capacitor gap is xcap,0 = 280 nm the maximum displacement given by the capacitive stiction point is δxnb ~ 110 nm, clearly larger than the initial nanobeam gap. Actuation is therefore not limited by the capacitive stiction point but by the nanobeam gap. Tuning as a function of the DC voltage for three different spring constants (kM = 0.1,1.0 and 10 N/m) is calculated and plotted in Fig. 5(a). A very large tuning range of Δλ = 75 nm can be achieved before the nanobeams touch. As in the all-optical case, the thermal Brownian motion at room temperature experienced by the nanobeams [Fig. 5(a) inset] results in wavelength fluctuation of the laser cavity. For the kM = 0.1 N/m MEMS device the wavelength noise standard deviation is σλ = 450 pm at maximum tuning. When compared to the all optical case, the cavity can be tuned farther but is accompanied by larger noise due to the fact that there is no dynamical spring effect to reduce the thermal fluctuations of the nanobeams.

Fig. 4. (a) The MEMS actuation system can be described as a RLC circuit coupled to a spring-mass system, where the coupling coefficient gEM is proportional to the change in the MEMS capacitance. This coupling strength can be describe in the same way of optical system, gEM = dΩLC/dxcap. (b) FEM simulation of the capacitance for the MEMS design showed in (c) as a function of the capacitor gap (xcap). On the right the evaluated force per applied voltage is also shown. The negative sign accounts for the attractive source of the force. (d) The two mechanical motion created by the push and pull capacitor.

As the electrostatic actuation produces a very large displacement, one could potentially increase the initial nanobeam gap to reduce the effects of thermal Brownian motion on laser cavity linewidth. Starting with a larger gap also eases fabrication tolerances. As a comparison, Fig. 5(b) shows the case where the initial beam gap is xnb,0 = 150 nm (capacitor gap size xcap,0 = 350 nm). The total tuning range, before the capacitor stiction point, is still significant at Δλ ~ 50 nm, but with roughly half the room temperature linewidth noise (σλ = 260 pm) for kM = 0.1 N/m.

In an electrical analogy to the all-optical case one can apply an AC field to actuate the MEMS. By solving the equation of motion for the capacitor plate coupled to the RLC charge equation we can write for the gap change:

Δx=4gEMkMΩLC2QLCη02×(Vac/2)2R,
(6)
Fig. 5. (a) Optical tuning as a function of the applied DC voltage for spring constants between 0.1 and 10.0 N/m. The initial gap between beams was taken to be 80 nm (capacitor gap 280 nm). (b) Same as (a) but for a initial gap of 150 nm between beams capacitor gap 350 nm. The insets show the resulting standard deviation of the cavity wavelength due to thermal noise (room temperature) as a function of the applied voltage for both cases.

where Vac is the amplitude of the monochromatic voltage source at frequency ω0, gEM = dΩLC/dxcap = (1/2)gCM/QLC, QLC is the RLC circuit quality factor, and the last term can be identified as the mean dropped power into the system. ∣η02 = Ω4LC/[(Ω2LC - ω20)22LCω20] is a dimensionless parameter with ΩLC/2π = 1/√LC, and ΓLC/2π = R/L being the RLC resonance frequency and damping rate respectively.

As expected the RLC circuit acts as a frequency band pass filter for the driving voltage source resulting in three different cases. First, for ω0≪ΩLC, ∣η02 ≈1 which results in a displacement similar to the DC case, where now the DC voltage is replaced by the time average voltage Vac /√2. Second, for ω0 ≫ ΩLC, ∣η02 ≈ 0 resulting in negligible displacement. The third case, when ω0 is close to the circuit’s resonance frequency, is an analog to the all-optical case where the RLC resonant circuit acts as the master cavity. In this case ∣η02 ≈Ω2LC/(4Δ22LC), where Δ = ω0LC is the detuning of the input voltage from the RLC resonance. The maximum change in the gap size is reached when Δ = 0 which leads to a frequency shift for the optical mode give by:

Δωs=gOM(s)4gEMQLCkMΩLC2×(Vac/2)2R.
(7)

Comparing this equation with Eq. (4) for the DC case, the presence of the RLC circuit is explicit. The actuation voltage is now inversely proportional to the quality factor of the circuit.

We can now evaluate the tuning curve of the optical mode as a function of the input voltage. This is shown in Fig. 6(a) for an illustrative set of parameters, ΩLC/2π = ω0/2π = 5 GHz and QLC = 100. Following the DC case, the initial beam gap size was taken as xnb,0 = 150 nm (capacitor gap size of xcap,0 = 350 nm) which gives the same total tuning range of Δλ ~ 50 nm.

Fig. 6. (a) Optical tuning as a function of the applied AC voltage for spring constants between 0.1 and 10.0 N/m for an AC signal at ωac/2π = 5 GHz. The RLC circuit was assumed to have a quality factor of QLC = 100. (b) The solid (dashed) lines shows the change in the mechanical spring constant ΔkM = kM(V = Vac)-kM(V = 0) as a function of the voltage for the optimal (no detuning) detuning from the RLC resonance frequency. The inset shows the fraction change of the spring constant ΔkM/kM as one detune the input voltage from the RLC resonance frequency. The dashed black line shows the resonance of the RLC circuit.

Due to the quality factor of the resonator the applied voltage needed is reduced. For the kM = 0.1 N/m device we need less than 40 mV to actuate and tune the structure through the full range.

The thermal Brownian motion at room temperature coupled to the optical cavity is the same as that of the DC case. It broadens the kM = 0.1 N/m device linewidth to σλ ~ 260 pm as seen in Fig. 6(b). Since Eq. (7) is essentially the same as Eq. (1), but for the electrical domain [26

26. V. B. Braginsky, F. Y. Khalili, and K. S. Thorne, Quantum Measurement. Cambridge University Press, (1995).

], one would expect an electric phenomena similar to the optical spring. In this case the detuned AC field modifies the mechanical spring constant. When the mechanical frequency is much smaller than the RLC resonance frequency we can evaluate the change of the spring constant as a function of the detuning Δ as:

kM=kM+2gEM2QLC[Δ(Δ2+(ΓLC/2)2)2]×(Vac/2)2R.
(8)

The inset on Fig. 6(b) shows the ratio of mechanical spring constant change ΔkM/kM as a function of the detuning for the maximum allowed voltage (before stiction point). Since we are in the side band unresolved regime the change in the spring constant does not depend on its initial value. The solid lines on Fig. 6(b) show the standard deviation of the cavity position resulting from thermal noise at the optimal detuning. Compared to when no detuning is applied (dashed lines) we can observe a reduction of the thermally-induced laser wavelength noise from σλ = 260 pm to ~ 130 pm for the kM = 0.1 N/m device at the full range. While in this work we have focused on the reduction of noise via stiffening of the mechanical structure by the optical or electrical spring effects, from the inset in Fig. 6(b) and Eqs. (3) and (8), one can see that it is possible to decrease stiffness as well.

4. Photoluminescence spectroscopy of zipper cavities

Here we present optical mode spectroscopy of zipper optomechanical cavities formed in light-emitting GaAs material. The active region in the GaAs material consists of embedded, self-assembled dots-in-a-well (DWELL) which emit at λ ≈ 1.2 μm [35

35. G. T. Liu, A. Stintz, H. Li, T. C. Newell, A. L. Gray, P. M. Varangis, K. J. Malloy, and L. F. Lester, “The influence of quantum-well composition on the performance of quantum dot lasers using InAs/InGaAs dots-in-a-well (DWELL) structures,” IEEE J. Quantum. Electron. 36, 1272–1279 (2000). [CrossRef]

, 36

36. A. Stintz, G. T. Liu, H. Li, L. F. Lester, and K. J. Malloy, “Low-threshold current density 1.3-μm InAs quantum-dot lasers with the dots-in-a-well (DWELL) structure,” IEEE Photon. Technol. Lett. 12, 591–593 (2000). [CrossRef]

]. Zipper cavities are fabricated by first depositing a hard silicon nitride (SiNx) mask ontop of a 255 nm thick GaAs membrane layer which contains the self-assembled InAs quantum dots (QDs). The cavity is patterned via electron-beam lithography, inductively-coupled reactive-ion (ICP-RIE) etching of the SiNx mask based on C4F8/SF6 chemistry, and subsequent pattern transfer into the QD membrane via ICP-RIE etching based on Ar/Cl2 chemistry. Devices are released from the substrate by wet etching of an underlying sacrificial AlGaAs layer in a dilute hydrofluoric acid solution (60 : 1 H2O:HF) [37

37. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006). [CrossRef] [PubMed]

]. Native oxide residue was removed using a dilute ammonium hydroxide solution (25 : 1 H2O:NH4OH) [38

38. J.W. Seo, T. Koker, S. Agarwala, and I. Adesida, “Etching characteristics of AlxGa1-xAs in (NH4)Sx solutions,” Appl. Phys. Lett. 60, 1114–1116 (1992). [CrossRef]

].

Fig. 7. (a) Photoluminescence experimental setup. (b) SEM micrograph top view of InAs/GaAs zipper cavity.

Devices are tested in a free space photoluminescence (PL) setup shown in Fig. 7(a). The devices, SEM images of which are shown in Fig. 7(b), are free space pumped with an external cavity diode laser (ECDL) at λ = 830 nm and free space photoluminescence is collected via a high numerical aperture objective and sent to a spectrometer coupled to an InGaAs charge-coupled device (CCD) array. The pump laser is pulsed to reduced thermal broadening with a period of τ = 1 μs, pulse width of w = 200 ns, resulting in an average pump power of Pp = 317 μW. A band pass filter is used to remove the residual reflected pump laser from the PL. A power meter at one port of a beam splitter measures the average incident power. The sample is mounted on top of a xyz-stage enabling rapid testing of multiple devices.

PL measurements show both the bonded and anti-bonded pairs as predicted and measured in passive cavities [17

17. J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystaloptomechanical cavity,” Opt. Express 17, 3802–3817 (2009). [CrossRef] [PubMed]

]. First order (red shade) through third order (yellow shade) defect cavity modes can be seen in Fig. 8(a). The modes collectively tune with increase of the photonic crystal lattice constant of the cavity as predicted by simulations. The wavelength splitting between bonded and anti-bonded modes is measured to be Δλ±= 5-7 nm. Such a splitting indicates a nanobeam gap of roughly 100 nm, consistent with the SEM images of the fabricated devices.

Figure 8(b) shows both bonded and anti-bonded modes at very low average pump power (w = 10 ns, τ = 4 μs, Pp = 101 nW). At this power level, and for the single layer of quantum dots in these devices, the expected enhancement/reduction to the pumped cavity Q-factor is negligible. From the linewidth of the modes, we estimate the quality factor of the bonded mode in these devices to be around Q ≈ 6000. Although not large enough to achieve lasing in this QD material (a Q-factor of greater than 5×104 is required to reach threshold given the limited QD gain), these results are encouraging given the much more substantial gain attainable in quantum well material [39

39. W.-Y. Hwang, J. Baillargeon, S. N. G. Chu, P. F. Sciortino, and A. Y. Cho, “GaInAsP/InP distributed feedback lasres grown directly on grated substrates by solid-source molecular beam epitaxy,” J. Vac. S. Tech. B 16, 1422–1425 (1998). [CrossRef]

].

Fig. 8. (a) Optically pumped zipper cavity modes spectra as a function of increasing a/λ. (b) Modes come in (+) bonded and anti bonded (-) pairs.

5. Discussion

In the all optical case, tuning range may be limited by the stiction point and therefore the initial nanobeam gap. In the case of tuning with reduced noise a tunable master laser source is required. Depending upon the geometry and application, there may also be issues associated with the absorption of the master laser light field, and its contribution to optical pumping of the slave cavity and the resulting generated free carriers. Not withstanding these potential shortcomings, we have shown that for reasonable master laser dropped power levels of 50 μW- 10 mW, a large radiation pressure force tuning of the slave cavity can be achieved up to a range of Δλ = 25 nm. Lower power actuation is attained by increasing the Q-factor or decreasing the static spring constant. The latter is associated with an increase in thermo-mechanical laser frequency noise. We have shown that a self-compensation of the thermal noise occurs in the optical tuning case due to dynamical back-action of the light field on the mechanical system via the optical spring effect. Even for the lowest static spring kM = 0.1 N/m considered in this work, the thermo-mechanical wavelength noise is reduced to below σλ ~ 11 pm over the full tuning range.

Analysis of a more conventional electrical-optical scheme is also performed to compare with the all-optical scheme. Two main advantages of the electro-optical scheme are the high magnitude of force available from electrostatic actuation, even at large gap distances, and the simple electrical control mechanism. In the laser structures studied in this work, the DC electrical tuning mechanism allows for as much as 75 nm of wavelength tuning or 5% of the nominal laser frequency. A drawback of the DC tuning mechanism is the inability to mediate thermal noise, except through an increase of the stiffness of the mechanical structure which in turn reduces the tuning range for a given applied voltage. The use of an AC voltage does allow for the increase of dynamic spring constant through electrical spring phenomena; however, the reduction of noise via the electrical spring is significantly less than in the optical case primarily because of the larger, attainable Q-factor in the optical domain. The Q of an RLC circuit can potentially be substantially increased by moving to superconducting materials at low temperature [40

40. P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, and J. Zmuidzinas, “A broadband superconducting detector suitable for use in large arrays,” Nature 425, 817–821 (2003). [CrossRef] [PubMed]

], although going to such means removes many of the practical advantages of the electro-optical scheme.

Numerous trade-offs exist between the all-optical and electrical-optical schemes. Although not studied here, one may envision a hybridized approach in which coarse laser frequency tuning is performed via electrostatic actuation, and fine laser frequency tuning and laser frequency stabilization are performed by an optical control field. There are also other interesting laser phenomena in addition to simple laser tuning that may be explored in similar optomechanical laser systems. The tuning dependence upon the master laser dropped power, and thus master laser frequency, in the all-optical case may be used as a frequency lock of master and slave lasers. The consideration of radiation pressure effects resulting from internal radiation, as in the case of the slave laser, leads to a whole new set of nonlinear dynamical equations and effects. Such effects include self-frequency-locking and mode-locking [41

41. H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000). [CrossRef]

]. Future work will aim to theoretically and experimentally explore these new dimensions to active cavity optomechanical systems.

Acknowledgments

The authors would like to thank Qiang Lin and Jessie Rosenberg for rich discussions and Jasper Chan for help with simulations. This work was supported by the DARPA NACHOS program (Award No. W911NF-07-1-0277).

References and links

1.

T. M. Hard, “Laser wavelength selection and output coupling by a grating,” Appl. Opt. 9, 1825–1830 (1970). [CrossRef] [PubMed]

2.

K. Liu and M. G. Littman, “Novel geometry for single-mode scanning of tunable lasers,” Opt. Express 6, 117–118 (1981).

3.

F. J. Duarte, Tunable Laser Applications, Second Edition. CRC, 2 ed. (2008). [CrossRef]

4.

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers. New York, NY: Van Nostrand Reinhold, (1993).

5.

K. Srinivasan and O. Painter, “Linear and nonlinear optical spectroscopy of a strongly coupled microdisk-quantum dot system,” Nature 450, 862–865 (2007). [CrossRef] [PubMed]

6.

E. Bruce, “Tunable Lasers,” IEEE Spectrum 39, 35–39 (2002). [CrossRef]

7.

A. Q. Liu and X. M. Zhang, “A review of MEMS external-cavity tunable lasers,” J. Micromech. Microeng. 17, R1–R13 (2007). [CrossRef]

8.

L. Coldren and S. Corzine, “Continuously-tunable single-frequency semiconductor lasers,” IEEE J. Quantum Electron. 23, 903–908 (1987). [CrossRef]

9.

Y. Matsui, H. Murai, S. Arahira, S. Kutsuzawa, and Y. Ogawa, “30-GHz bandwidth 1.55 μm strain-compensated InGaAlAs-InGaAsP MQW laser,” IEEE Photonics Technol. Lett. 9, 25–27 (1997). [CrossRef]

10.

M. Kitamura, M. Seki, M. Yamaguchi, I. Mito, K. Kobayashi, and T. Matsuoka, “High-power single-longitudinal-mode operation of 1.3 μm DFB-DC-PBH LD,” Electron. Lett. 19, 840–841 (1983). [CrossRef]

11.

L. Diehl, B. G. Lee, P. Behroozi, M. Loncar, M. A. Belkin, F. Capasso, T. Aellen, D. Hofstetter, M. Beck, and J. Faist, “Microfluidic tuning of distributed feedback quantum cascade lasers,” Opt. Express 14, 11660–11667 (2006). [CrossRef] [PubMed]

12.

V. Moreau, R. Colombelli, R. Perahia, O. Painter, L. R. Wilson, and A. B. Krysa, “Proof-of-principle of surface detection with air-guided quantum cascade lasers,” Opt. Express 16, 8387–6396 (2008). [CrossRef]

13.

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nature Photon. 2, 180–184 (2008). [CrossRef]

14.

M.-C. M. Lee and M. C. Wu, “MEMS-Actuated microdisk resonators with variable power coupling ratios,” IEEE Photon. Technol. Lett. 17, 1034–1036 (2005). [CrossRef]

15.

K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, “Cooling and control of a cavity optoelectromechanical system,” arXiv:0909.0082v3 [quant-ph], (2009).

16.

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-Action at the mesoscale,” Science 321, 1172–1176 (2008). [CrossRef] [PubMed]

17.

J. Chan, M. Eichenfield, R. Camacho, and O. Painter, “Optical and mechanical design of a “zipper” photonic crystaloptomechanical cavity,” Opt. Express 17, 3802–3817 (2009). [CrossRef] [PubMed]

18.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009). [CrossRef] [PubMed]

19.

R. M. Camacho, J. Chan, M. Eichenfield, and O. Painter, “Characterization of radiation pressure and thermal effects in a nanoscale optomechanical cavity,” Opt. Express 17, 15726–15735 (2009). [CrossRef] [PubMed]

20.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (2002). [CrossRef]

21.

M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature 462, 78–82 (2009). [CrossRef] [PubMed]

22.

N. Tas, T. Sonnenberg, H. Jansen, R. Legtenberg, and M. Elwenspoek, “Stiction in surface micromachining,” J. Micromech. Microeng. 6, 385–397 (1996). [CrossRef]

23.

R. Maboudian and R. T. Howe, “Critical review: Adhesion in surface micromechanical structures,” J. Vac. Sci. Technol. B 15, 1–20 (1997). [CrossRef]

24.

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nature Photon. 3, 478–483 (2009). [CrossRef]

25.

T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express 15, 17172–17205 (2007). [CrossRef] [PubMed]

26.

V. B. Braginsky, F. Y. Khalili, and K. S. Thorne, Quantum Measurement. Cambridge University Press, (1995).

27.

Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009). [CrossRef] [PubMed]

28.

R. Perahia, T. P. M. Alegre, A. H. Safavi-Naeini, and O. Painter, “Surface-plasmon mode hybridization in subwavelength microdisk lasers,” Appl. Phys. Lett. 95, 201114 (2009). [CrossRef]

29.

M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. Veldhoven, F. W. M. V. Otten, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2007). [CrossRef]

30.

P. R. Rice and H. J. Carmichael, “Photon statistics of a cavity-QED laser: A comment on the laser-phase transition analogy,” Phys. Rev. A 50, 4318–4329 (1994). [CrossRef] [PubMed]

31.

G. Björk, A. Karlsson, and Y. Yamamoto, “On the line width of lasers,” Appl. Phys. Lett. 60, 304–306 (1992).

32.

R. E. Slusher, “Optical processes in microcavities,” Semicond. Sci. Technol. 9, 2025–2030 (1994). [CrossRef]

33.

I. W. Frank, P. B. Deotare, M. W. McCutcheon, and M. Loncar, “Dynamically reconfigurable photonic crystal nanobeam cavities,” arXiv:0909.2278, (2009).

34.

M. A. Sillanpaa, J. Sarkar, J. Sulkko, J. Muhonen, and P. J. Hakonen, “Accessing nanomechanical resonators via a fast microwave circuit,” Appl. Phys. Lett. 95, 011909 (2009). [CrossRef]

35.

G. T. Liu, A. Stintz, H. Li, T. C. Newell, A. L. Gray, P. M. Varangis, K. J. Malloy, and L. F. Lester, “The influence of quantum-well composition on the performance of quantum dot lasers using InAs/InGaAs dots-in-a-well (DWELL) structures,” IEEE J. Quantum. Electron. 36, 1272–1279 (2000). [CrossRef]

36.

A. Stintz, G. T. Liu, H. Li, L. F. Lester, and K. J. Malloy, “Low-threshold current density 1.3-μm InAs quantum-dot lasers with the dots-in-a-well (DWELL) structure,” IEEE Photon. Technol. Lett. 12, 591–593 (2000). [CrossRef]

37.

K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, “Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots,” Opt. Express 14, 1094–1105 (2006). [CrossRef] [PubMed]

38.

J.W. Seo, T. Koker, S. Agarwala, and I. Adesida, “Etching characteristics of AlxGa1-xAs in (NH4)Sx solutions,” Appl. Phys. Lett. 60, 1114–1116 (1992). [CrossRef]

39.

W.-Y. Hwang, J. Baillargeon, S. N. G. Chu, P. F. Sciortino, and A. Y. Cho, “GaInAsP/InP distributed feedback lasres grown directly on grated substrates by solid-source molecular beam epitaxy,” J. Vac. S. Tech. B 16, 1422–1425 (1998). [CrossRef]

40.

P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, and J. Zmuidzinas, “A broadband superconducting detector suitable for use in large arrays,” Nature 425, 817–821 (2003). [CrossRef] [PubMed]

41.

H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000). [CrossRef]

OCIS Codes
(220.4880) Optical design and fabrication : Optomechanics
(230.4685) Optical devices : Optical microelectromechanical devices
(250.5960) Optoelectronics : Semiconductor lasers

ToC Category:
Optoelectronics

History
Original Manuscript: December 23, 2009
Revised Manuscript: March 26, 2010
Manuscript Accepted: March 27, 2010
Published: March 31, 2010

Citation
Thiago P. Mayer Alegre, Raviv Perahia, and Oskar Painter, "Optomechanical zipper cavity lasers: theoretical analysis of tuning range and stability," Opt. Express 18, 7872-7885 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-7872


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References

  1. T. M. Hard, "Laser wavelength selection and output coupling by a grating," Appl. Opt. 9, 1825-1830 (1970). [CrossRef] [PubMed]
  2. K. Liu and M. G. Littman, "Novel geometry for single-mode scanning of tunable lasers," Opt. Express 6, 117-118 (1981).
  3. F. J. Duarte, Tunable Laser Applications, Second Edition. CRC, 2 ed. (2008). [CrossRef]
  4. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers. New York, NY: Van Nostrand Reinhold, (1993).
  5. K. Srinivasan and O. Painter, "Linear and nonlinear optical spectroscopy of a strongly coupled micro disk quantum dot system," Nature 450, 862-865 (2007). [CrossRef] [PubMed]
  6. E. Bruce, "Tunable Lasers," IEEE Spectrum 39, 35-39 (2002). [CrossRef]
  7. A. Q. Liu and X. M. Zhang, "A review of MEMS external-cavity tunable lasers," J. Micromech. Microeng. 17, R1-R13 (2007). [CrossRef]
  8. L. Coldren and S. Corzine, "Continuously-tunable single-frequency semiconductor lasers," IEEE J. Quantum Electron. 23, 903-908 (1987). [CrossRef]
  9. Y. Matsui, H. Murai, S. Arahira, S. Kutsuzawa, and Y. Ogawa, "30-GHz bandwidth 1.55 μm strain-compensated InGaAlAs-InGaAsP MQW laser," IEEE Photonics Technol. Lett. 9, 25-27 (1997). [CrossRef]
  10. M. Kitamura, M. Seki, M. Yamaguchi, I. Mito, K. Kobayashi, and T. Matsuoka, "High-power single-longitudinal mode operation of 1.3 μm DFB-DC-PBH LD," Electron. Lett. 19, 840-841 (1983). [CrossRef]
  11. L. Diehl, B. G. Lee, P. Behroozi, M. Loncar, M. A. Belkin, F. Capasso, T. Aellen, D. Hofstetter, M. Beck, and J. Faist, "Microfluidic tuning of distributed feedback quantum cascade lasers," Opt. Express 14, 11660-11667 (2006). [CrossRef] [PubMed]
  12. V. Moreau, R. Colombelli, R. Perahia, O. Painter, L. R. Wilson, and A. B. Krysa, "Proof-of-principle of surface detection with air-guided quantum cascade lasers," Opt. Express 16, 8387-8396 (2008). [CrossRef]
  13. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, "A nanoelectromechanical tunable laser," Nature Photon. 2, 180-184 (2008). [CrossRef]
  14. M.-C. M. Lee and M. C. Wu, "MEMS-Actuated microdisk resonators with variable power coupling ratios," IEEE Photon. Technol. Lett. 17, 1034-1036 (2005). [CrossRef]
  15. K. H. Lee, T. G. McRae, G. I. Harris, J. Knittel, and W. P. Bowen, "Cooling and control of a cavity optoelectromechanical system," arXiv:0909.0082v3 [quant-ph], (2009).
  16. T. J. Kippenberg and K. J. Vahala, "Cavity optomechanics: Back-Action at the mesoscale," Science 321, 1172-1176 (2008). [CrossRef] [PubMed]
  17. J. Chan, M. Eichenfield, R. Camacho, and O. Painter, "Optical and mechanical design of a "zipper" photonic crystaloptomechanical cavity," Opt. Express 17, 3802-3817 (2009). [CrossRef] [PubMed]
  18. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, "A picogram- and nanometre-scale photoniccrystal optomechanical cavity," Nature 459, 550-555 (2009). [CrossRef] [PubMed]
  19. R. M. Camacho, J. Chan, M. Eichenfield, and O. Painter, "Characterization of radiation pressure and thermal effects in a nanoscale optomechanical cavity," Opt. Express 17, 15726-15735 (2009). [CrossRef] [PubMed]
  20. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002). [CrossRef]
  21. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, "Optomechanical crystals," Nature 462, 78-82 (2009). [CrossRef] [PubMed]
  22. N. Tas, T. Sonnenberg, H. Jansen, R. Legtenberg, and M. Elwenspoek, "Stiction in surface micromachining," J. Micromech. Microeng. 6, 385-397 (1996). [CrossRef]
  23. R. Maboudian and R. T. Howe, "Critical review: Adhesion in surface micromechanical structures," J. Vac. Sci. Technol. B 15, 1-20 (1997). [CrossRef]
  24. J. Rosenberg, Q. Lin, and O. Painter, "Static and dynamic wavelength routing via the gradient optical force," Nature Photon. 3, 478-483 (2009). [CrossRef]
  25. T. J. Kippenberg and K. J. Vahala, "Cavity Opto-Mechanics," Opt. Express 15, 17172-17205 (2007). [CrossRef] [PubMed]
  26. V. B. Braginsky, F. Y. Khalili, and K. S. Thorne, Quantum Measurement. Cambridge University Press, (1995).
  27. Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, "Mechanical oscillation and cooling actuated by the optical gradient force," Phys. Rev. Lett. 103, 103601 (2009). [CrossRef] [PubMed]
  28. R. Perahia, T. P. M. Alegre, A. H. Safavi-Naeini, and O. Painter, "Surface-plasmon mode hybridization in subwavelength microdisk lasers," Appl. Phys. Lett. 95, 201114 (2009). [CrossRef]
  29. M. T. Hill, Y. S. Oei, B. Smalbrugge, Y. Zhu, T. D. Vries, P. J. Veldhoven, F. W. M. V. Otten, J. P. Turkiewicz, H. D. Waardt, E. J. Geluk, S. H. Kwon, Y. H. Lee, R. Notzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2007). [CrossRef]
  30. P. R. Rice and H. J. Carmichael, "Photon statistics of a cavity-QED laser: A comment on the laser-phase transition analogy," Phys. Rev. A 50, 4318-4329 (1994). [CrossRef] [PubMed]
  31. G. Björk, A. Karlsson, and Y. Yamamoto, "On the line width of lasers," Appl. Phys. Lett. 60, 304-306 (1992).
  32. R. E. Slusher, "Optical processes in microcavities," Semicond. Sci. Technol. 9, 2025-2030 (1994). [CrossRef]
  33. I. W. Frank, P. B. Deotare, M. W. McCutcheon, and M. Loncar, "Dynamically reconfigurable photonic crystal nanobeam cavities," arXiv:0909.2278, (2009).
  34. M. A. Sillanpaa, J. Sarkar, J. Sulkko, J. Muhonen, and P. J. Hakonen, "Accessing nanomechanical resonators via a fast microwave circuit," Appl. Phys. Lett. 95, 011909 (2009). [CrossRef]
  35. G. T. Liu, A. Stintz, H. Li, T. C. Newell, A. L. Gray, P. M. Varangis, K. J. Malloy, and L. F. Lester, "The influence of quantum-well composition on the performance of quantum dot lasers using InAs/InGaAs dots-in-awell (DWELL) structures," IEEE J. Quantum. Electron. 36, 1272-1279 (2000). [CrossRef]
  36. A. Stintz, G. T. Liu, H. Li, L. F. Lester, and K. J. Malloy, "Low-threshold current density 1.3-μm InAs quantum dot lasers with the dots-in-a-well (DWELL) structure," IEEE Photon. Technol. Lett. 12, 591-593 (2000). [CrossRef]
  37. K. Srinivasan, M. Borselli, O. Painter, A. Stintz, and S. Krishna, "Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots," Opt. Express 14, 1094-1105 (2006). [CrossRef] [PubMed]
  38. J.W. Seo, T. Koker, S. Agarwala, and I. Adesida, "Etching characteristics of AlxGa1−xAs in (NH4)Sx solutions," Appl. Phys. Lett. 60, 1114-1116 (1992). [CrossRef]
  39. W.-Y. Hwang, J. Baillargeon, S. N. G. Chu, P. F. Sciortino, and A. Y. Cho, "GaInAsP/InP distributed feedback lasers grown directly on grated substrates by solid-source molecular beam epitaxy," J. Vac. S. Tech. B 16, 1422-1425 (1998). [CrossRef]
  40. P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, and J. Zmuidzinas, "A broadband superconducting detector suitable for use in large arrays," Nature 425, 817-821 (2003). [CrossRef] [PubMed]
  41. H. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000). [CrossRef]

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